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Counterexample to regularity in averagedistance problem
 OF MINIMIZERS OF AVERAGEDISTANCE PROBLEM 21
, 2012
"... The averagedistance problem is to find the best way to approximate (or represent) a given measure µ on Rd by a onedimensional object. In the penalized form the problem can be stated as follows: given a finite, compactly supported, positive Borel measure µ, minimize E(Σ) = d(x, Σ)dµ(x) + λH 1 (Σ ..."
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Cited by 7 (2 self)
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The averagedistance problem is to find the best way to approximate (or represent) a given measure µ on Rd by a onedimensional object. In the penalized form the problem can be stated as follows: given a finite, compactly supported, positive Borel measure µ, minimize E(Σ) = d(x, Σ)dµ(x) + λH 1
AVERAGEDISTANCE PROBLEM FOR PARAMETERIZED CURVES
"... ABSTRACT. We consider approximating a measure by a parameterized curve subject to length penalization. That is for a given finite compactly supported measure µ, with µ(Rd)> 0 for p ≥ 1 and λ> 0 we consider the functional E(γ) = Rd d(x,Γγ) pdµ(x) + λLength(γ) where γ: I → Rd, I is an interval i ..."
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in R, Γγ = γ(I), and d(x,Γγ) is the distance of x to Γγ. The problem is closely related to the averagedistance problem, where the admissible class are the connected sets of finite Hausdorff measure H1, and to (regularized) principal curves studied in statistics. We obtain regularity of minimizers
More counterexamples to regularity for minimizers of the average distance problem
"... The averagedistance problem, in the penalized formulation, involves minimizing; (1) Eλµ(Σ):= d(x,Σ)dµ(x) + λH1(Σ), among pathwise connected, closed sets Σ with finite H1measure, where d ≥ 2, µ is a given measure, λ is a given parameter and d(x,Σ): = infy∈Σ x − y. The averagedistance problem ca ..."
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Cited by 1 (1 self)
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The averagedistance problem, in the penalized formulation, involves minimizing; (1) Eλµ(Σ):= d(x,Σ)dµ(x) + λH1(Σ), among pathwise connected, closed sets Σ with finite H1measure, where d ≥ 2, µ is a given measure, λ is a given parameter and d(x,Σ): = infy∈Σ x − y. The averagedistance problem
REGULARITY OF DENSITIES IN RELAXED AND PENALIZED AVERAGE DISTANCE PROBLEM
"... ABSTRACT. The average distance problem finds application in data parameterization, which involves “representing ” the data using lower dimensional objects. From a computational point of view it is often convenient to restrict the unknown to the family of parameterized curves. However this formulatio ..."
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ABSTRACT. The average distance problem finds application in data parameterization, which involves “representing ” the data using lower dimensional objects. From a computational point of view it is often convenient to restrict the unknown to the family of parameterized curves. However
EXAMPLE OF MINIMIZER OF THE AVERAGEDISTANCE PROBLEM WITH NON CLOSED SET OF CORNERS
, 2014
"... The averagedistance problem, in the penalized formulation, involves minimizing; Eλµ(Σ):= inf y∈Σ x − ydµ(x) + λH1(Σ), among pathwise connected, closed sets Σ with finite H1measure, where d ≥ 2, µ is a given measure and λ a given parameter. Regularity of minimizers is a delicate problem. It is ..."
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The averagedistance problem, in the penalized formulation, involves minimizing; Eλµ(Σ):= inf y∈Σ x − ydµ(x) + λH1(Σ), among pathwise connected, closed sets Σ with finite H1measure, where d ≥ 2, µ is a given measure and λ a given parameter. Regularity of minimizers is a delicate problem
PROPERTIES OF MINIMIZERS OF AVERAGEDISTANCE PROBLEM VIA DISCRETE APPROXIMATION OF MEASURES
"... ABSTRACT. Given a finite measure µ with compact support, and λ> 0, the averagedistance problem, in the penalized formulation, is to minimize (0.1) E λ ∫ µ(Σ): = d(x, Σ)dµ(x) + λH 1 (Σ), Rd among pathwise connected, closed sets, Σ. Here d(x, Σ) is the distance from a point to a set and H1 is the 1 ..."
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Cited by 1 (0 self)
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ABSTRACT. Given a finite measure µ with compact support, and λ> 0, the averagedistance problem, in the penalized formulation, is to minimize (0.1) E λ ∫ µ(Σ): = d(x, Σ)dµ(x) + λH 1 (Σ), Rd among pathwise connected, closed sets, Σ. Here d(x, Σ) is the distance from a point to a set and H1
Where the REALLY Hard Problems Are
 IN J. MYLOPOULOS AND R. REITER (EDS.), PROCEEDINGS OF 12TH INTERNATIONAL JOINT CONFERENCE ON AI (IJCAI91),VOLUME 1
, 1991
"... It is well known that for many NPcomplete problems, such as KSat, etc., typical cases are easy to solve; so that computationally hard cases must be rare (assuming P != NP). This paper shows that NPcomplete problems can be summarized by at least one "order parameter", and that the hard p ..."
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Cited by 681 (1 self)
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It is well known that for many NPcomplete problems, such as KSat, etc., typical cases are easy to solve; so that computationally hard cases must be rare (assuming P != NP). This paper shows that NPcomplete problems can be summarized by at least one "order parameter", and that the hard
Cognitive load during problem solving: effects on learning
 COGNITIVE SCIENCE
, 1988
"... Considerable evidence indicates that domain specific knowledge in the form of schemes is the primary factor distinguishing experts from novices in problemsolving skill. Evidence that conventional problemsolving activity is not effective in schema acquisition is also accumulating. It is suggested t ..."
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Cited by 603 (13 self)
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Considerable evidence indicates that domain specific knowledge in the form of schemes is the primary factor distinguishing experts from novices in problemsolving skill. Evidence that conventional problemsolving activity is not effective in schema acquisition is also accumulating. It is suggested
Theoretical improvements in algorithmic efficiency for network flow problems

, 1972
"... This paper presents new algorithms for the maximum flow problem, the Hitchcock transportation problem, and the general minimumcost flow problem. Upper bounds on ... the numbers of steps in these algorithms are derived, and are shown to compale favorably with upper bounds on the numbers of steps req ..."
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Cited by 565 (0 self)
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This paper presents new algorithms for the maximum flow problem, the Hitchcock transportation problem, and the general minimumcost flow problem. Upper bounds on ... the numbers of steps in these algorithms are derived, and are shown to compale favorably with upper bounds on the numbers of steps
Nonlinear component analysis as a kernel eigenvalue problem

, 1996
"... We describe a new method for performing a nonlinear form of Principal Component Analysis. By the use of integral operator kernel functions, we can efficiently compute principal components in highdimensional feature spaces, related to input space by some nonlinear map; for instance the space of all ..."
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Cited by 1554 (85 self)
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We describe a new method for performing a nonlinear form of Principal Component Analysis. By the use of integral operator kernel functions, we can efficiently compute principal components in highdimensional feature spaces, related to input space by some nonlinear map; for instance the space of all possible 5pixel products in 16x16 images. We give the derivation of the method, along with a discussion of other techniques which can be made nonlinear with the kernel approach; and present first experimental results on nonlinear feature extraction for pattern recognition.
Results 1  10
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